Two routes from the
Boltzmann equation to
compressible flow of
polyatomic gases
Paul J. Dellar
Department of Mathematics,
Imperial College London,
London SW7 2AZ,
United Kingdom
Email: [email protected]
Abstract: This paper presents a systematic approach to simulating compressible flow
of polyatomic gases using the Boltzmann equation for a discrete set of particle velocities. We derive the complete system of moment equations needed to recover the
Navier–Stokes–Fourier equations. One may either circumvent the usual relation between pressure and internal energy density by assigning additional energies to the
particles, or introduce an entirely separate set of particle distribution functions to simulate the macroscopic energy equation. The latter permits the use of longer timesteps,
and may generalise more easily to multiple space dimensions. However, the momentum
and energy equations must be coupled to obtain correct viscous heating for realistic
values of the Prandtl number. Numerical experiments are presented for the standard
one dimensional Sod shock tube benchmark for monatomic and diatomic gases using
both unified 7 velocity and split 4+3 velocity formulations.
Keywords: lattice Boltzmann; discrete kinetic theory; gas dynamics; shock waves;
splitting methods
Reference to this paper should be made as follows: Dellar, P. J. (2008) ‘Two routes
from the Boltzmann equation to compressible flow of polyatomic gases’, Progress in
Computational Fluid Dynamics Vol. 8, Nos. 1/2/3/4 pp. 84–96.
DOI:10.1504/PCFD.2008.018081
Biographical notes: Paul Dellar graduated in mathematics from the University of
Cambridge, and holds a PhD in applied mathematics from the same institution. At
the time of writing he held a Lectureship in Applied Mathematics at Imperial College
London, but he has since moved to the University of Oxford.
1
Introduction
discretisation employed in the lattice Boltzmann method,
rather than any intrinsic defect in Boltzmann equation
itself. The Boltzmann equation with a discrete velocity
space, equation (1) below, is just a finite set of hyperbolic
partial differential equations coupled by some algebraic
source terms. Using alternative space/time discretisations
designed for hyperbolic systems (for pioneering work see
Refs. [22, 25]), it is possible to simulate standard benchmark problems such as the first Sod [20, 32] shock tube
using a discrete velocity Boltzmann equation [9].
The lattice Boltzmann method is now widely employed
to simulate nearly incompressible fluid flows, most commonly using an isothermal equation. Efforts to simulate
compressible flows with evolving temperatures at larger
Mach numbers originally met with limited success, being
far more susceptible to numerical instabilities [1, 5]. This
led to a search for alternative equilibria offering improvements in numerical stability [26], but at the price of introducing unphysical artifacts such as spikes and compound
Before embarking on a study of Boltzmann-based methwaves into the solutions of the simulated equations [7].
ods for compressible flow, it is worth remarking that the
However, it is now recognised that the lack of numerical stability is a consequence of the particular space/time
c
Copyright ° 2008 Published by Inderscience Enterprises Ltd.
1
more established total variation diminishing (TVD) or
Godunov-based methods (see e.g. Laney [20]) only attempt to recover entropy-satisfying solutions of the compressible Euler equations, without explicit treatment of
non-ideal effects like viscosity and heat conduction. Moreover, these methods are known to malfunction in various
scenarios due to a lack of sufficient dissipation, perhaps the
most well-known being the spurious transonic rarefaction
shocks computed by Roe’s approximate Riemann solver
[24, 27]. To take another example, a Godunov-based computation in magnetohydrodynamics showed incorrect behaviour due to an inaccurate treatment of dissipation in
the magnetic field, as described in [8].
Xu’s [39] gas kinetic BGK scheme offers an integrated
treatment that combines the ideal Euler equations with
viscosity and heat conduction. However, this approach
reconstructs the distribution function as a truncated
Chapman–Enskog expansion, with gradients of the velocity
and temperature approximated using their values at grid
points. In other words, the only degrees of freedom are the
standard hydrodynamic variables of mass, momentum, and
energy densities at grid points. A discrete or lattice Boltzmann approach provides independent evolution equations
for the stress, heat flux, and higher moments of the distribution function. This may offer an improved approximation to continuum kinetic theory outside the small meanfree-path limit described by the Chapman–Enskog expansion, and motivates existing applications of the standard
lattice Boltzmann method to rarefied flows of monatomic
gases at small Mach numbers. These are currently confined to isothermal flows in relatively simple geometries,
but a number of lattice Boltzmann implementations have
been demonstrated that capture, at least qualitatively, second order slip flow and the so-called Knudsen minimum in
microchannels [2, 35, 40]. However, a more satisfactory
treatment would employ larger lattices in the discrete velocity space, chosen to reproduce the higher moments of
the continuum Maxwell–Boltzmann distribution function
that enter the derivation of the Burnett equations at second order in Knudsen number. The present paper is itself
concerned with the adoption of larger lattices in discrete
velocity space, but for the different purpose of simulating
the correct energy equation for a polyatomic gas.
Turning to polyatomic gases, in any ideal gas the pressure p varies according to p ∝ ργ under adiabatic changes
in the density ρ. Kinetic theory gives the relation γ =
1 + 2/D between the adiabatic exponent γ and the number D of degrees of freedom associated with molecule. This
leads to the well-known γ = 5/3 for monatomic gases, each
atom having 3 translational degrees of freedom. Much recent interest has been devoted to simulating the behaviour
of gases in micro-electro-mechanical devices (MEMs). The
gas is typically air, primarily a mixture of two diatomic
gases nitrogen and oxygen. Diatomic molecules have 2 rotational degrees of freedom, rotation about the line of centres being forbidden quantum-mechanically, which combines with the 3 translational degrees of freedom to give
D = 5 and γ = 7/5. The commercial heat transfer fluid
known as PP10, the fluorocarbon C13 F22 , has γ = 1.0128
corresponding to D ≈ 39. (Equality is not exact because
the molecule comprises carbon and fluorine atoms with two
different masses.)
Being based on the kinetic theory of monatomic gases,
those with no internal degrees of freedom, the value for D
in the standard lattice or discrete Boltzmann approach is
just the number of spatial dimensions. Thus one obtains
D = 3 and γ = 5/3 from a three dimensional computation, but γ = 3 in the purely one-dimensional formulation
used in Ref. [9]. As it would be undesirable to formulate a lattice in 5, let alone 39, spatial dimensions merely
to simulate flow of one of the above fluids, we shall investigate approaches that allow the adiabatic exponent γ
to be varied independently of the number of spatial dimensions. Following earlier work on the classical kinetic
theory of polyatomic gases [12, 23, 34], Shi et al. [30] and
Kataoka and Tsutahara [14] equipped their particles with
additional internal energies to circumvent the restrictions
of the Boltzmann equation. However, Shi et al. [30] did
not attempt to obtain correct viscous and conductive behaviour, being satisfied with the compressible Euler equations, while Kataoka and Tsutahara [14] postulate a form
(0)
for their equilibrium distribution functions fi that typically leads to an unnecessarily large number of particle
velocities.
Li et al. [21] developed a discrete Boltzmann approach
for simulating aeroacoustics in diatomic gases. They retained the unmodified Boltzmann equation without internal energies, as presented in Sec. 2 below. Being concerned
only with simulating small amplitude sound waves (acoustics) they adjusted the second moment of the equilibrium
distribution to give the correct energy density for diatomic
gases. The subsequently incorrect momentum flux tensor
does not affect the propagation of sound waves, so this
approach is perfectly sufficient for the purpose of simulating aeroacoustics. It is not, however, sufficient to simulate
more general flows of diatomic gases. Li et al. [21] also implemented a temperature-dependent relaxation time that
yields Sutherland’s law for the temperature-dependent viscosity of a diatomic gas (see Sec. 10).
In this paper we give a systematic treatment, first deriving necessary and sufficient conditions to recover the
Navier–Stokes–Fourier equations exactly, then investigating two alternative implementations. One may either use
a single set of particle to reproduce the continuity, momentum, and energy equations, or introduce a second set
of particles to reproduce only the energy equation. The
latter may be more amenable to implementation in multiple space dimensions, but it is necessary to couple the two
sets of particles through the collision operator to obtain
the correct viscous heating when the Prandtl number is
set to a realistic value.
Our approach is valid in the limit of a rapid equipartition of energy between the translational and rotational
degrees of freedom, for which the gas may be described
by the Navier–Stokes–Fourier equations with an arbitrary
(but constant) adiabatic exponent γ. A more sophisti-
2
(0)
cated treatment with a separate evolution equation for
the internal energy becomes necessary when the timescale
for exchange of energy between translational and rotational degrees of freedom is comparable to hydrodynamic
timescales. At the other extreme, when hydrodynamic
timescales are short compared with the timescale for internal energy exchange, the internal degrees of freedom
become “frozen” and the gas behaves as though it were
monatomic (e.g. [16]). In addition, real gas molecules
contain different kinds of internal degrees of freedom, or
modes. For example, a diatomic molecule has a vibrational mode in addition to the rotational modes described
above. The vibrational mode is typically frozen at room
temperature, but may become excited at higher temperatures. This leads to an effective change in the adiabatic
exponent γ with temperature, as each additional internal
mode moves from being frozen, through evolving on a hydrodynamic timescale, to a rapid equipartition with the
translational degrees of freedom. The approach in this paper is limited to constant γ.
constraints on the functional form of the fi expressed in
terms of the quantities ρ, u, and E. These constraints ensure that mass, momentum, and energy are conserved under the BGK model of collisions, just as they are conserved
by Boltzmann’s original binary collision operator. Requir(0)
ing the existence of a solution fi to these constraints, and
of the further constraints derived below, in turn imposes
constraints on the ξ i in a more subtle manner.
From moments of the Boltzmann equation (1) we obtain
exact conservation laws for the macroscopic mass, momentum, and energy densities,
∂t ρ + ∇·(ρu) = 0,
∂t E + ∇·F = 0.
(4)
The right hand sides vanish due to equations (3). The
momentum flux (or stress) tensor Π and the energy flux
vector F are identified as higher moments of the distribution functions,
Π=
N
X
∂t (ρu) + ∇·Π = 0,
N
ξi ξi fi ,
F=
i=0
2
The discrete Boltzmann equation is an evolution equation
for a finite set of distribution functions fi (x, t),
Π(0)
N
X
fi ,
ρu =
i=0
N
X
(1)
i=0
E=
i=0
(0)
= ρ,
N
X
i=0
(0)
ξ i fi
1X
|ξ |2 fi ,
2 i=0 i
(2)
=
1X
(0)
|ξ |2 ξ f = u(E + p),
2 i=0 i i i
Π = Π(0) + ²Π(1) + · · · ,
(6)
(7)
F = F (0) + ²F (1) + · · · , (9)
where ² is the ratio of τ to a characteristic hydrodynamic timescale. The conserved quantities ρ, u, and E are
left unexpanded, since they do not evolve on a collisional
timescale. One then uses the corresponding multiple-scales
expansion of the time derivative
N
= ρu,
= pI + ρuu,
and using a Chapman–Enskog expansion to seek solutions
that vary slowly compared with the collisional timescale τ .
This is equivalent to expanding the momentum and energy
fluxes as series in a small parameter ²,
(0)
fi
(0)
ξ i ξ i fi
where p is the fluid pressure. These equations should thus
be interpreted as imposing additional constraints on the
(0)
equilibria fi . An expression for the pressure p in terms
of the conserved quantities ρ, u, E will be given in the
following section.
The Navier–Stokes–Fourier equations follow by formulating evolution equations for the higher moments, for instance
ÃN
!
´
X
1³
ξ i ξ i ξ i fi = −
∂t Π + ∇·
Π − Π(0) ,
(8)
τ
i=0
where u is the fluid velocity. The equilibria fi are then
constructed as functions of these macroscopic quantities in
such a way that
N
X
N
X
N
F (0)
N
ξi fi ,
=
i=0
Each fi represents the number density of particles moving
with velocity ξ i . On the right hand side we have written
the Bhatnagar, Gross, Krook (BGK) collision operator [3],
under which the fi relax towards some explicitly specified
(0)
equilibria fi with a single relaxation time τ . The choice
of a BGK collision operator, and a fixed τ , is adopted
for simplicity of presentation, and will be briefly revisited
later. The common objection that the BGK collision operator yields suboptimal stability with the space/time discretisation of (1) used in the lattice Boltzmann method is
not relevant for the alternative space/time discretisation
that we describe in Sec. 9 below.
The macroscopic mass (ρ), momentum (ρu), and energy
(E) densities are typically defined by moments of the distribution functions,
ρ=
(5)
The compressible Euler equations follow from (4) when Π
and F are approximated by their equilibrium values, which
must be
Moments of the Boltzmann equation
´
1³
(0)
∂t fi + ξi · ∇fi = −
fi − fi
.
τ
1X
|ξ |2 ξ fi .
2 i=0 i i
1X
(0)
|ξ |2 f = E. (3)
2 i=0 i i
∂t = ∂t0 + ²∂t1 + · · ·
(10)
to justify replacing ∂t by ∂t0 in equation (8) and its analogue for F . Further details are presented in Sec. 4 for the
momentum flux, and Sec. 5 for the energy flux.
To clarify, the former equations (2) are definitions of the
quantities ρ, u, and E, while the latter equations (3) are
3
3
Introduction of internal energies
temperature θ such that p = θρ. This temperature is measured in so-called energy units that absorb Boltzmann’s
constant, or the molar gas constant R. The temperature
is then equal to the square of the isothermal or Newtonian
sound speed, cs = (p/ρ)1/2 = θ1/2 .
An alternative definition of temperature for a polyatomic gas uses only the translational degrees of freedom
[16]. This alternative definition leads to a ∇·u term in
the relation between pressure and temperature, in place of
the bulk viscous stress proportional to ∇·u that we calculate below. For example, Kogan’s [15] treatment using the
translational temperature yields exactly the same expression −τ ρθ(5/3 − γ)∇·u that we give in (23) below, but as
a non-equilibrium contribution to the pressure rather than
an isotropic contribution to the viscous stress from bulk
viscosity.
Comparing equations (3) and (7), we see that there is a
relation between the energy density E and the trace of the
equilibrium momentum flux tensor Π(0) ,
E=
N
N
1X
1 X
1
(0)
(0)
|ξi |2 fi = Tr
ξi ξi fi = Tr Π(0) . (11)
2 i=0
2
2
i=0
Substituting Π(0) = pI + ρuu as given by the compressible
Euler equations, we obtain
E=
D
1
p + ρ|u|2 ,
2
2
(12)
where the number of spatial dimensions D enters through
Tr I = D. Comparing equation (12) with the standard
expression for an ideal gas [19]
E=
p
1
+ ρ|u|2 ,
γ−1 2
4
(13)
Following the standard Chapman–Enskog approach, we
obtain an explicit formula for the viscous stress from the
evolution equation (8) by evaluating the left hand side us(0)
ing the equilibrium distributions fi , and approximating
the time derivative using the compressible Euler equations,
!
ÃN
X
1
(0)
(0)
∂t0 Π + ∇·
ξi ξi ξi fi
= − Π(1) .
(18)
τ
i=0
establishes the relation γ = 1 + 2/D between D and the
adiabatic exponent γ.
Following earlier work on the classical kinetic theory of
polyatomic gases [12, 23, 34], Shi et al. [30] and Kataoka
and Tsutahara [14] circumvented this restriction on γ by
postulating an additional internal energy ²i for each particle, in addition to the particle’s kinetic energy proportional
to |ξi |2 . They thus replaced the previous definitions of the
macroscopic energy density and energy flux by
E=
1
2
N
X
i=0
¡ 2
¢
|ξi | + ²i fi ,
F=
1
2
N
X
Calculation of the viscous stress
The first term ∂t0 Π(0) = ∂t0 (ρuu + ρθI) evaluates to
¡ 2
¢
|ξi | + ²i ξi fi .
∂t0 Π(0)
i=0
(14)
Having postulated the expression for E, the expression for
F follows from comparing the desired energy equation
=
=
=
∂t E + ∇·F = 0
(15)
∂t0 (ρu)u + u∂t0 (ρu) − uu∂t0 ρ + ∂t0 (ρθ)I,
−∇·(ρuu + ρθI)u − u∇·(ρuu + ρθI)
+ uu∇·(ρu) − ∇·(ρθu)I − (γ − 1)(ρθ∇·u)I,
−∇·(ρuuu) − ∇(ρθ)u − u∇(ρθ)
−∇·(ρθu)I − (γ − 1)(ρθ∇·u)I,
(19)
with the Boltzmann equation (1). Note that the ²i must be after using the Euler momentum and temperature equaprescribed constants, so that they may be taken inside spa- tions in the form
tial and temporal derivatives to construct evolution equations for the moments.
∂t0 (ρu) + ∇·(ρuu + ρθI) = 0,
To obtain the compressible Euler equations at leading
∂t0 (ρθ) + ∇·(ρθu) + (γ − 1)ρθ∇·u = 0.
(20)
order, we require [19]
To cancel the ρuuu and ∇(ρθ) terms in (19) we must
(0)
choose the equilibrium distribution fi to satisfy
N
X¡
¢
1
1
p
(0)
+ ρ|u|2 ,
(16)
|ξi |2 + ²i fi =
E (0) =
N
2 i=0
γ−1 2
X
(0)
ξ i ξ i ξ i fi = ρuuu + ρθ (u I + cyclic) .
(21)
N
X
¡
¢
1
1
γ
(0)
(0)
2
2
i=0
F
=
|ξi | + ²i ξi fi =
pu + ρ|u| u.
2 i=0
γ−1
2
The expression (u I + cyclic) denotes the completely sym(17) metric third rank tensor with components uα δβγ +uβ δγα +
uγ δαβ that cannot be written in dyadic notation. SubstiThese equations should be interpreted as constraints on tuting (19) and (21) into (18), we obtain
(0)
both the fi and the internal energies ²i .
¶
µ
For future convenience, and following standard termi∂uβ
∂uγ
∂uα
(1)
+
− (γ − 1)δαβ
,
(22)
Παβ = −τ ρθ
nology in the lattice Boltzmann literature, we introduce a
∂xβ
∂xα
∂xγ
4
which is a Newtonian viscous stress with shear viscosity and
µ = τ ρθ. It may be rewritten in a more illuminating form
¢
¡
as
∂t0 12 ρu|u|2
1 2
·
¸
µ
¶
=
|u| ∂t0 (ρu) + u u · ∂t0 (ρu) − u|u|2 ∂t0 ρ,
2
5
(1)
T
2
Π = −τ ρθ (∇u) + (∇u) − I∇·u −τ ρθ
− γ I∇·u.
3
3
1
= − |u|2 ∇·(ρuu + ρθI),
(23)
2
The first term, having zero trace, is the standard viscous
−u u · ∇·(ρuu + ρθI) + u|u|2 ∇·(ρu),
stress for a dilute monatomic gas (for which γ = 5/3) ob1
1
tained from classical kinetic theory. The second term is an
= − ∇·(ρuu|u|2 ) − |u|2 ∇(ρθ) − uu · ∇(ρθ),
2 ·
2
¸
additional bulk viscous stress with bulk viscosity (5/3−γ)µ
1
1
2
2
that arises from changing the adiabatic exponent γ. This
= −∇· ρuu|u| + |u| ρθI + ρθuu ,
2
2
definition of bulk viscosity is always positive when γ < 5/3,
1
corresponding to additional internal degrees of freedom
+ρθ∇ |u|2 + ρθ∇·(uu).
(27)
2
(D > 3 in γ = 1 + 2/D), but different authors use different
conventions for what constitutes “bulk viscosity”. Some
authors use this phrase for the whole of the isotropic term Assembling the two halves,
in (22), rather than just that part adding to the traceless
∂t0 F (0)
first term in (23), as listed in [6].
¶
µ
1
γ
2
In summary, the constraints (3), (7), and (21) on the
ρu|u| +
ρθu
= ∂t0
(0)
2
γ−1
equilibrium distribution functions fi are necessary and
·
¸
1
1 2
2γ − 1
γ
sufficient to yield the continuity and momentum equations
2
2
= ∇· ρuu|u| + |u| ρθI +
ρθuu +
ρθ I
2
2
γ−1
γ−1
with a Newtonian viscous stress as above. The only use we
γ
have made of the energy equation is through eliminating
+ρθ [uα ∇uα + u · ∇u − (γ − 1)u∇·u] +
ρθ∇θ,
∂t0 (ρθ) using the compressible Euler temperature equation
γ−1
·
¸
to derive (19).
1
1
2γ − 1
γ
= ∇· ρuu|u|2 + |u|2 ρθI +
ρθuu +
ρθ2 I
2
2
γ−1
γ−1
1
γ
− u · Π(1) +
ρθ∇θ.
(28)
τ
γ−1
5
Calculation of heat conduction and viscous heating
(0)
Choosing the fi so that the term whose divergence
appears in (25) cancels the divergence term ∇·[ ] in (28),
Similarly, we calculate the corrections to the energy flux
F (0) that are responsible for heat conduction and viscous
heating by first formulating an evolution equation for F .
Taking the 12 (|ξ i |2 + ²i )ξi moment of the Boltzmann equation gives
Ã
N
¢
1
1
1 X¡ 2
(0)
|ξi | + ²i ξi ξi fi = ρ|u|2 uu + ρθ|u|2 I
2 i=0
2
2
+
!
´
1³
1
∂t F + ∇·
(|ξ i |2 + ²i ) ξi ξi fi = −
F − F (0) ,
2 i=0
τ
(24)
which, to sufficient accuracy, again simplifies to
N
X
Ã
∂t0 F (0) +∇·
when using the
∂t0 F (0) step by
∂t0 (ρθu)
=
=
=
1
2
N
X
!
(0)
(|ξ i |2 + ²i ) ξ i ξ i fi
2γ − 1
γ
ρθuu +
ρθ2 I,
γ−1
γ−1
(29)
therefore gives the desired viscous heating and conductive
heat flux,
F (1) = u · Π(1) −
1
= − F (1) (25)
τ
γ
τ ρθ∇θ.
γ−1
(30)
The first term is the rate of working by viscous stresses,
while the second term gives Fourier’s law with thermal
conductivity κ = τ ρθγ/(γ − 1).
Chapman–Enskog expansion. Evaluating
Given that the right hand side of (29) must be a second
step, we find
rank symmetric tensor with the dimensions of ρ|u|4 , a reasonable alternative to the above calculation is to postulate
u∂t0 (ρθ) + θ∂t0 (ρu) − uθ∂t0 ρ,
the functional form of (29) with four arbitrary constants
in place of 1/2, γ/(γ − 1), etc. Evaluating the left hand
−u [∇·(ρuθ) + (γ − 1)ρθ∇·u]
side of (25) using a symbolic manipulator package, the four
−θ∇·(ρuu + ρθI) + uθ∇·(ρu),
constants may be adjusted to eliminate any appearance of
−(γ − 1)ρθu∇·u − θ∇(ρθ) − ∇·(ρθuu),
∇ρ in the heat flux. This procedure yields the unique co(26) efficients given in (29).
i=0
5
6
where we note that the two O(|u|2 ) terms arise from the
contraction on two indices of the completely symmetric
fourth rank tensor with components uα uβ δγδ + · · · as required by the structure of the left hand side.
Constraints on the equilibria and internal energies
To summarise, the equations of mass and momentum conservation with a Newtonian viscous stress determine the
moments
N
X
(0)
fi
=
N
X
ρ,
i=0
Π
(0)
ξ i fi
The γ-dependent contribution that remains for the ²i
terms is then
= ρu,
i=0
(0)
N
X
=
(0)
ξ i ξ i fi
N
= ρθI + ρuu,
1X
(0)
|ξ |2 ²i fi =
2 i=0 i
(31)
i=0
N
X
(0)
ξ i ξ i ξ i fi
=
µ
1
D
−
γ−1
2
¶
¡
¢
ρθuu + ρθ2 I ,
(37)
ρuuu + ρθ (u I + cyclic) .
i=0
which establishes the pattern of relations
The compressible Euler energy equation determines
N
¢ (0)
1 X¡ 2
|ξi | + ²i fi
2 i=0
=
N
¢
1 X¡ 2
(0)
|ξi | + ²i ξi fi
2 i=0
=
ρθ
1
+ ρ|u|2 ,
γ−1 2
(32)
1
γ
ρ θ u + ρ|u|2 u,
γ−1
2
N
1X
2
and the conductive heat flux and viscous heating together
determine
+
(33)
2γ − 1
γ
ρθuu +
ρθ2 I.
γ−1
γ−1
N
=
D
1
ρθ + ρ|u|2 ,
2
2
=
1
2+D
ρ|u|2 u +
ρθu,
2
2
N
1X
(0)
|ξ |2 ξ f
2 i=0 i i i
so to satisfy equations (32) in addition we need
¶
µ
N
1X
D
1
(0)
−
ρθ,
²i f
=
2 i=0 i
γ−1
2
¶
µ
N
1X
D
1
(0)
−
ρθu,
² i ξ i fi
=
2 i=0
γ−1
2
=
µ
=
i=0
N
=
D
1
−
γ−1
2
¶ X
N
(0)
θ
fi ,
i=0
¶ X
N
1
D
(0)
−
θ
ξi fi , (38)
γ−1
2
i=0
µ
¶ X
N
1
D
(0)
−
θ
ξi ξi fi ,
γ−1
2
i=0
between the ²i moments and the lower moments appearing in the continuity and momentum equations. Unfortunately, taking the ²i proportional to θ as suggested by this
pattern is not possible. The ²i must be constants, so that
they commute with spatial and temporal derivatives, to
obtain moment equations in conservation form as above.
By contracting equations (31) on two indices, we find
that the ²i -independent contributions to (32) are already
determined to be
1X
(0)
|ξ |2 f
2 i=0 i i
(0)
²i ξi fi
1X
(0)
²i ξi ξi fi
2 i=0
N
¢
1
1
1 X¡ 2
(0)
|ξi | + ²i ξ i ξ i fi = ρ|u|2 uu + ρθ|u|2 I
2 i=0
2
2
µ
N
1X
(0)
²i f
2 i=0 i
(34)
7
One dimensional, unsplit seven velocity model
Each constraint equation has only one component in one
spatial dimension. There are thus four constraints (31)
from the momentum equation, and another three, (32) and
where D is the number of spatial dimensions as before.
(33), from the energy equation. This suggests choosing
Since the fourth moment does not enter the momentum seven velocities with ξi = i for i = −3, . . . , 3. The ²i must
equation at viscous order, there is no unique way to split be chosen so that {²i , ξi ²i , ξ 2 ²i } and {1, ξi , ξ 2 , ξ 3 } comprise
i
i
i
(33) between the |ξi |2 and ²i contributions. However, it seven linearly independent vectors in R7 . These vectors are
might seem reasonable to adopt the γ = 5/3 value for the rows of the 7 × 7 matrix appearing below. One such
the |ξi |2 contribution, so that the ²i are responsible for all choice is ²i = (0, 0, 1, 4, 1, 0, 0), motivated by the threedeviations from monatomic gas behaviour. This gives
velocity isothermal lattice Boltzmann equilibria. The perhaps more natural rescaling ²i = (0, 0, 1/6, 2/3, 1/6, 0, 0)
N
X
1
7
5
1
1
(0)
|ξi |2 ξ i ξ i fi = ρ|u|2 uu+ ρθ|u|2 I+ ρθuu+ ρθ2 I, leads to a far less stable numerical scheme.
2 i=0
2
2
2
2
(36)
Given the ²i , the constraints (31) to (33) may be written
(35)
6
(0)
model using an entirely separate set of distribution functions to evolve the fluid’s internal energy. They used
a standard D2Q9 lattice for each distribution function,
which meant that there were no particles with speed 2
or greater to cause instability, but also that their momentum equation contained artifacts due to the omission of the
ρuuu term in the third moment of the distribution function for density and momentum, our equation (21). This
omission restricted the validity of their scheme to thermal
flows at small Mach numbers, although they retained various terms such as viscous heating that would normally be
neglected in a Boussinesq limit. As a side effect, use of two
independent distribution functions, each with a BGK collision operator, enabled He et al. [13] to adjust the Prandtl
number by varying the two relaxation times independently.
The mass and momentum conservation equations, including the Newtonian viscous stress, arise from equations
(31) that do not involve the ²i . One may therefore choose
Subtracting the third row from the fifth row, and the fourth the ξ and f (0) to satisfy just the continuity and momeni
i
row from the sixth row, leads to some simplification,
tum equations, and obtain the complete energy equation


from an entirely separate set of distribution functions gi

 f (0)
−3
obeying a second discrete Boltzmann equation
1
1
1 1 1 1 1  (0) 
 −3 −2 −1 0 1 2 3  f−2 

  (0) 
1
(0)
 9


∂t gi + ζ i · ∇gi = − (gi − gi ),
(42)
4
1 0 1 4
9

 f−1

τ
(0)
−27 −8 −1 0 1 8 27 f 

 0 
 0
 (0) 
with a possibly different set of discrete particle velocities
0
1 4 1 0
0

 f1 
 0
ζ i . The two discrete Boltzmann equations for fi and gi are
(0) 
0 −1 0 1 0 0  
f2 
coupled
via the moments ρ, u, and θ evaluated at lattice
81
16
2 0 2 16 81
(0)
f3
points.
Achieving
a realistic Prandtl number also requires


coupling the equation for the gi to the viscous stress com1
(0)


u
puted from fi − fi .


2
(0)


u +θ
The first few moments of the equilibria gi must be


3
 . (40)
u
+
3θu
= ρ
given by




θ(γ − 3)/(γ − 1)


N


θu(γ − 3)/(γ − 1)
1 X (0)
ρθ
1
gi
=
+ ρ|u|2 ,
(43a)
4
2
2
u + θu (5γ − 3)/(γ − 1) + 2θ γ/(γ − 1)
2 i=0
γ−1 2
as a system of seven linear equations for the seven fi ,



 f (0)
−3
1
1
1 1 1 1
1  (0) 
 −3 −2 −1 0 1 2 3  f−2 

  (0) 
 9


4
1 0 1 4
9

 f−1 
−27 −8 −1 0 1 8 27 f (0) 

 0 
 9
 (0) 
4
2 4 2 4
9

 f 
−27 −8 −2 0 2 8 27  1(0) 
f2 
81
16
2 0 2 16 81
(0)
f3


1


u


2


u
+
θ


3
 . (39)
u
+
3θu
= ρ


2


u
+
2θ/(γ
−
1)




u3 + 2θu γ/(γ − 1)
u4 + θu2 (5γ − 3)/(γ − 1) + 2θ2 γ/(γ − 1)
Having chosen the ²i to ensure linear independence of the
rows of the matrices in (39) and (40), either system may
be solved using a symbolic manipulator to obtain seven
concrete expressions such as
µ
12γ − 62
ρ
(0)
−36 + 13|u|2 − |u|4 −
θ
f0 =
64
γ−1
¶
5γ − 3
2γ 2
−
θ|u|2 −
θ . (41)
γ−1
γ−1
N
1X
(0)
ζg
2 i=0 i i
γ
1
ρ θ u + ρ|u|2 u,
γ−1
2
=
1
1
ρ|u|2 uu + ρθ|u|2 I
2
2
N
1X
(0)
ζζg
2 i=0 i i i
+
(43b)
2γ − 1
γ
ρθuu +
ρθ2 I. (43c)
γ−1
γ−1
The left hand side of (43c) is a generic symmetric second
rank tensor, unlike equation (36) before, so the O(|u|2 )
terms need not be (and are not) the contraction of a symmetric fourth rank tensor.
Four momentum constraints in one dimension suggest
using four particles with velocities ξi = {−2, −1, 1, 2}. In
a departure from convention, there are no rest particles in
this set of velocities. As before, the equilibria are given by
the solution
¢
1 ¡ 3
(0)
ρ ±u + 2u2 ± (3θ − 1)u + 2θ − 2 ,
f±2 =
12
¢ (44)
1 ¡
(0)
f±1 = − ρ ±2u3 + 2u2 ± (6θ − 8)u + 2θ − 8 ,
12
It is usually more computationally efficient to write equilibria like this, one formula for each value of i, rather than as
one complicated expression involving weights wi and the ξi
that is valid for i = −3, . . . 3. One typically gains efficiency
from unrolling the resulting loop over i and precomputing
the functions of ξi as constants.
8
=
One dimensional, split 4+3 velocity model
The poor stability properties of the early thermal lattice
Boltzmann models led He et al. [13] to develop a split
7
∆t using the trapezium rule [13]. However, although equation (49) is second order accurate for ∆t ¿ τ , equation
(0)
(49) causes the nonequilibrium part f i − fi to oscillate
at every timestep when τ < 12 ∆t. By contrast, the original
(0)
partial differential equation (1) always causes fi − fi to
decay monotonically towards zero on a timescale τ .
Moreover, when using particles with speeds 2 and larger,
as needed to reproduce thermal behaviour, some information propagates 2 or more grid points per timestep under (49). Although the advective part of (49) is neutrally stable, consisting only of shifting data between grid
points, it would not be surprising if the coupled system became unstable through violating the spirit of the Courant–
Friedrichs–Lewy (CFL) stability condition – that information should propagate no further than one grid spacing per
timestep.
This heuristic argument is not the whole story, because
one isothermal D2Q13 scheme is stable enough to perform
useful computations [38], and linearly stable D1Q5 schemes
may be found for general barotropic equations of state [10].
However, Lallemand and Luo [18] found that the thermal
D2Q13 scheme is linearly unstable due to an interaction between the energy flux and the shear stress. They proposed
using a conventional finite difference method for the energy equation instead, but their finding of instability only
applies to the standard space/time discretisation (49).
We therefore adopt an alternative space/time discretisation that does not suffer from either of these possible
sources of instability. Using Strang splitting [33] we combine a solution of the advection equation ∂t fi +ξ i ·∇fi = 0
from the Beam–Warming method [20, 37] with an exact solution of the collision equation. This splitting procedure
resembles the “spin steps” used to implement the Coriolis
force in a lattice Boltzmann ocean model [28]. The solution finew to the discrete Boltzmann equation (1) after
a timestep ∆t is given by three steps, with intermediate
values fi? and fi?? ,
µ
¶
1 ∆t
(0)
(0)
?
fi = fi + (fi − fi ) exp −
,
(51a)
2 τ
µ
1
∆t
3fi? (x) − 4fi? (x − ∆x)
fi?? (x) = fi? (x) − |ξi |
2
∆x
¶
?
+fi (x − 2∆x)
to the four linear equations (31), rewritten in matrix form
as


  (0) 

f−2
1
1 1 1
1
(0) 
−2 −1 1 2 


f 
u


.
(45)
 −1
 = ρ
2
(0)
4


1 1 4 f1 
u +θ 
(0)
−8 −1 1 8
u3 + 3θu
f
2
If one were to include an additional rest particle, there
would be five equilibria to be found, but still only four
constraint equations. This leaves one undetermined degree
of freedom that may be adjusted to optimise stability [10].
The separate energy equation requires three constraints
(43a-c), suggesting a separate set of three particle velocities
ζi = {−1, 0, 1} that does include a rest particle to maintain
symmetry. Writing the three constraint equations (43) in
matrix form gives
  (0) 
g−1
1 1 1
(0) 
−1 0 1 
g0 
(0)
1 0 1
g1


u2 + 2θ/(γ − 1)

u3 + 2θuγ/(γ − 1)
= ρ
4
2
u + θu (5γ − 3)/(γ − 1) + 2θ2 γ/(γ − 1)
 
S
= V  . (46)
T

The symbols S, V , T denote scalar, vector, and tensor quantities of the right hand side respectively (though
only the x and xx components are present in this onedimensional formulation). This particularly simple system
has the solution
(0)
g−1 =
9
1
(T − V ) ,
2
(0)
g0 = S − T,
(0)
g+1 =
1
(T + V ) .
2
(47)
Numerical implementation
The standard lattice Boltzmann method approximates the
system of partial differential equations
´
1³
(0)
fi − fi
(48)
∂t fi + ξ i · ∇fi = −
τ
by the set of algebraic equations
f i (x + ξ i ∆t, t + ∆t) − f i (x, t) = −
³
´
∆t
(0)
(f i − fi
.
τ + ∆t/2
(49)
(new)
fi
The f i are related to the fi by
=
1
2
µ
∆t
∆x
¶2 µ
fi? (x) − 2fi? (x − ∆x)
¶
+fi? (x − 2∆x) , (51b)
¶
µ
1 ∆t
(0)
(0)
.
(51c)
fi + (fi?? − fi ) exp −
2 τ
+
|ξi |
The Beam–Warming formula (51b) is written for ξi > 0,
and combines a second-order accurate upwind first difference for the advection with an upwind second difference for
which renders (49) a second order accurate and fully ex- stability. For ξi < 0 one should replace x − ∆x by x + ∆x,
plicit approximation to (1). Equation (49) may be derived replace x − 2∆x by x + 2∆x, and reverse the sign of the
by integrating (1) along characteristics for a time interval first term.
f i = fi +
´
∆t ³
(0)
fi − fi
,
2τ
(50)
8
(0)
the viscous heating correct, it is necessary to adopt a collision operator that is non-diagonal in the basis of moments
with respect to the particle velocity ξ i . In continuum kinetic theory there is no difficulty because it is standard to
use moments with respect to the so-called peculiar velocity c = ξi − u instead. This separates the u · Π(1) and
−κ∇θ contributions of the energy flux, so one may use
various extensions of the BGK collision operator such as
the Gross–Jackson [11] or Shakhov S-model [29] that allow
arbitrary Prandtl number, yet are diagonal in a basis of
moments with respect to the peculiar velocity. Transforming back to a basis of moment with respect to ξi yields a
non-diagonal collision operator, but one that can still be
solved exactly using exponentials. Further investigation,
and a detailed study of the splitting method, will presented
elsewhere.
In addition, a constant value for the collision time τ ,
adopted here for simplicity, gives a dynamic viscosity
µ = τ ρθ that is proportional to the product of density
and temperature. This is an artifact of the BGK collision
operator. The viscosity in a dilute monatomic gas is proportional to θ1/2 and independent of density. At fixed temperature, the longer mean free path in a lower density gas
precisely compensating for the smaller number of atoms
per unit volume transporting momentum. The temperature dependence for diatomic gases is commonly modelled
by Sutherland’s law,
¶3/2
µ
θref + θS
θ
.
(54)
µ = µref
θref
θ + θS
Since fi depends only on the quantities ρ, u and θ that
are conserved under collisions, steps (51a) and (51c) give
the exact solution to the system of ordinary differential
equations
1
(0)
∂t fi = − (fi − fi )
(52)
τ
over a time interval 21 ∆t. In particular, these steps ensure
(new)
(0)
that fi
→ fi as τ → 0 even when ∆t À τ . This socalled asymptotic preserving property ensures that small τ
values do not impose a stability constraint on the timestep
∆t. However, there is an accuracy constraint due to the
splitting of the coupled advection and collision equation
(1) into three separate steps. The use of two collision steps
(51a) and (51c), each for times 12 ∆t, gives a splitting error
of O(∆t2 ). The overall scheme thus has second-order accuracy in time. The two collision steps may be combined
into one for an efficient implementation, except when the
solution at a particular time n∆t for integer n is required
for output.
It is worth emphasising that the above split scheme leads
to accurate results when ∆t ≈ τ , with little benefit from
taking ∆t ¿ τ , but the splitting error leads to overly diffusive behaviour when ∆t À τ . Unlike the standard lattice
Boltzmann scheme (49), the split scheme does not capture
the correct behaviour of slowly varying solutions of the
discrete Boltzmann PDE (48) when ∆t À τ . This limitation is shared by other approximations to the discrete
Boltzmann PDE, such as finite volume approximations to
the isothermal discrete Boltzmann PDE on unstructured
meshes [36]. It is perhaps a less severe restriction for simulations of compressible flow with shocks, since substantially more dissipation is required to resolve shocks than
is required to resolve shear layers in incompressible flows.
The thickness of shocks is proportional to the viscosity µ,
while the thickness of shear layers is proportional to µ1/2 .
In other words, the resolution of shocks in a compressible
flow requires a grid-scale Reynolds number of order unity,
in the absence of some kind of solution-adaptive dissipation.
Here µref is the viscosity at a reference temperature θref ,
and θS is a constant (equivalent to 111 Kelvin for air).
Sutherland’s law asymptotes to the monatomic gas behaviour at high temperatures, µ ∼ θ1/2 when θ À θS .
A more accurate simulation would take the energy and
momentum relaxation times to be separate functions of
the density and temperature at lattice points, chosen to
give the desired functional dependence of the viscosity and
thermal conductivity on temperature (as in [21]).
10
11
Changing the Prandtl number
Figure 1 shows the results of simulating Sod’s first shock
tube problem [20, 32] using the unsplit seven velocity
model of Sec. 7 with γ = 5/3. The initial conditions for
this benchmark problem correspond to a stationary gas
with density and pressure given by
Using the unsplit approach of Sec. 7 with the BGK collision
operator leads to non-equilibrium momentum and energy
fluxes given by
£
¤
Π(1) = −µ (∇u) + (∇u)T −(γ −1)I∇·u ,
F (1)
= u · Π(1) − κ∇θ,
Numerical results
ρ = 1.0 and
ρ = 0.125 and
(53)
with viscosity µ = τ ρθ and thermal conductivity κ =
µγ/(γ − 1). This corresponds to a Prandtl number Pr =
(ν/κ) · (γ − 1)/γ of unity in standard kinetic theory usage,
but the true value of the Prandtl number should be close
to 2/3, and exactly 2/3 for a gas of Maxwell molecules.
To adjust the value of κ independently of the µ appearing inside Π(1) in the second of equations (53), thus leaving
p = 1.0 for x < 0,
p = 0.1 for x > 0.
(55)
The computation was performed on the domain −0.5 ≤
x < 0.5 using 4096 points and periodic boundary conditions. There is thus a second backwards-facing shock tube
at the periodic boundary, but the two Riemann fans have
not had time to interact by the time t = 0.1 shown in the
figure. The computations used a BGK collision operator
9
(0)
ΠNSF
xx − Πxx = −(4/3)τ ρθux ,
1
ρ
0.6
0.4
0.2
0
−0.2
1
There is no fixed relation between the temperature and the
particle velocities in discrete kinetic theory, since the dis-
10
0.2
0.1
0.2
Boltzmann
conventional
0.4
0.2
0
−0.2
−0.1
0
x
0.8
(57)
0.1
0.6
as computed from the numerical values of ρ, u, and θ, and
second order centred differences for their derivatives. The
actual non-equilibrium fluxes are in close agreement with
their Navier–Stokes–Fourier values, except in the region of
(0)
the contact discontinuity where Πxx − Πxx has the wrong
sign compared with the Navier–Stokes viscous stress. The
true solution should have spatially uniform velocity at the
contact discontinuity, which is marked by a jump in density
only, so the Navier–Stokes viscous stress should vanish.
The discrepancy may be due to finite Knudsen number
(finite τ ) effects, or a consequence of initialising with the
piecewise constant initial data (55).
Figures 3 and 4 show the corresponding data for a diatomic gas with γ = 7/5. The previous initial conditions
were scaled by a factor of 1/4 in the pressure to achieve a
stable computation,
p = 0.25 for x < 0,
p = 0.025 for x > 0.
0
0.8
1
ρ = 1.0 and
ρ = 0.125 and
−0.1
x
(56)
Boltzmann
conventional
0.6
p
FxNSF − Fx(0) = −(4/3)τ ρθuux − (5/2)τ ρθθx ,
Boltzmann
conventional
0.8
u
with τ = 10−4 , and a timestep ∆t set by the maximum
Courant number 3∆t/∆x = 0.9. For these parameter values the ratio ∆t/τ ≈ 0.8 is not especially small, but the solution was left virtually unchanged by taking much smaller
timesteps.
The solution computed with fixed τ is in close agreement
with a reference solution of the compressible Euler equations computed using the second order accurate extension
by Kurganov and Tadmor [17] of the local Lax–Friedrichs
(or Rusanov) scheme. The system of ordinary differential equations arising from this semi-discrete scheme was
solved using a second order, total variation diminishing,
Runge–Kutta integrator [31]. The discrete Boltzmann solution shows a small velocity overshoot due to insufficient
dissipation at the shock, and a velocity perturbation near
the contact discontinuity.
Total variation diminishing (TVD) methods for compu- (a)
tational gas dynamics capture shocks without overshoots,
and without excessive global dissipation, by increasing the
dissipation in the neighbourhood of large spatial gradients
(see e.g. Laney [20]). The so-called entropic lattice Boltzmann methods achieve a similar goal by adjusting the relaxation time depending upon the non-equilibrium part of
the distribution functions, which in turn depends upon local spatial gradients (see e.g. Boghosian et al. [4]). A
fully competitive Boltzmann scheme for polyatomic gases
is likely to require the implementation of an entropic approach to improve shock capturing in comparison with
computations using a uniform relaxation time as presented
here.
Figure 2 compares the non-equilibrium momentum and
(0)
(0)
energy fluxes Πxx − Πxx and Fx − Fx with their Navier–
(b)
Stokes–Fourier values,
0.4
0.2
(c)
0
−0.2
−0.1
0
0.1
0.2
x
Figure 1: Comparison of (a) the density, (b) the velocity, and (c) the pressure for Sod’s first shock tube problem
with γ = 5/3 at time t = 0.1. Results from the unsplit
Boltzmann scheme are shown solid, and a conventional numerical solution for the inviscid problem is shown dotted.
There is a small velocity overshoot at the shock, and a
bump at the contact discontinuity near x = 0.07.
locity near the contact discontinuity is no longer noticeable, compare Fig. 1(b) with Fig. 3(b), but discrepancies
in the non-equilibrium fluxes near the contact discontinuity are noticeably larger. The latter may be partly because
the magnitudes of the Navier–Stokes–Fourier viscous stress
and energy flux are roughly a factor of 8 smaller due to the
rescaling of temperature and velocity in (57).
The split 4+3 velocity model developed in Sec. 8 turns
out to be much less stable than the unsplit 7 velocity
model. Figure 5 shows a comparison of the density computed using both the split 4+3 velocity and unsplit 7 velocity models. However, even with γ = 5/3 and the rescaled
temperature, it was necessary to smooth out the initial
data using a tanh function so that the initial gradient was
no more than 50 in modulus. Both computations used a
grid with 4096 points, τ = 10−4 , and maximum Courant
number of 0.9. However, the split model uses 50% longer
timesteps because the maximum particle speed is 2, rather
than 3, in the CFL stability criterion.
−3
x 10
1
Boltzmann
Navier−Stokes
Πxx − Π(0)
xx
0.5
0
−0.5
−1
−0.2
(a)
−0.1
0
0.1
0.2
x
−3
x 10
1
Fx − F(0)
x
0
12
−1
We have derived a set of moment equations leading to
the compressible Navier–Stokes–Fourier equations with arbitrary adiabatic exponent γ, and a correct, Galileaninvariant viscous stress and energy flux. For general γ
1
the usual
P monatomic gas relations E = 2 Tr Π and F =
1
i ξ i ξ i ξ i fi ) do not hold, so it is necessary to in2 Tr (
troduce additional degrees of freedom to satisfy the constraints on E and F independently of thoseP
for the momentum flux Π and complete third moment i ξi ξi ξi fi .
One may either assign additional internal energies to
particles, allowing the moments of fi in the energy equation to be adjusted independently of those in the momentum equation, or introduce a completely separate set of
distribution functions gi for just the energy equation. In
either case, there are seven moment constraints in one spatial dimension, requiring at least seven different particle
velocities to satisfy them. By contrast, five particle velocities are sufficient to reproduce the Navier–Stokes–Fourier
equations with the natural γ = 3 for one-dimensional kinetic theory [9]. The unsplit 7 velocity model developed in
Sec. 7 stably simulates the first Sod shock tube problem for
both γ = 5/3 (monatomic) and γ = 7/5 (diatomic). The
split 4+3 velocity model is noticeably less stable, but allows the use of longer timesteps. However, the space/time
discretisation in Sec. 9 using Strang splitting and Beam–
Warming advection is just one of many possible numerical
implementations.
In discrete kinetic theory there is no fixed relation between the sound speed, as given by the temperature, and
the particle speeds. The temperature θ is a free parameter in all the discrete equilibrium distributions given in
this paper. The fluid velocity and the sound speed may
thus both be rescaled with respect to the particle speeds
without changing the Mach number, the ratio of the fluid
velocity to the sound speed. The finite particle speeds thus
−2
−3
−4
(b)
−5
−0.2
Boltzmann
Navier−Stokes
−0.1
0
0.1
Conclusion
0.2
x
Figure 2: Comparison of (a) the viscous stress, and (b) the
diffusive energy flux, for the same computation as Fig. 1.
The non-equilibrium parts of Πxx and Fx are shown solid,
and the Navier–Stokes–Fourier values for these quantities
obtained from the computed values of ρ, u, θ are shown
dotted. There is good agreement apart from the viscous
stress near the contact discontinuity, where the true solution should have spatially uniform velocity.
(0)
crete equilibria fi are not constrained by the functional
form of the continuum Maxwell–Boltzmann distribution
involving the ratio |ξ − u|2 /θ. Conversely, it is reasonable
to expect that some relation between the square root of
temperature, or isothermal sound speed, and the particle
velocities will be necessary for stability. For instance, stability would be highly unlikely if the sound speed exceeded
the fastest particle speed. One may recover the solution
of the original unscaled problem by suitably scaling the
fluid velocity, timescale, and transport coefficients in the
simulation.
Again, there is reasonable agreement between the discrete Boltzmann solution, performed on a grid of 8192
points with τ = 10−4 as before, and the reference solution of the inviscid equations. A perturbation in ve-
11
−4
x 10
1
Boltzmann
conventional
Boltzmann
Navier−Stokes
1
Πxx − Π(0)
xx
0.8
ρ
0.6
0.4
0
−1
0.2
0
−0.2
(a)
−0.1
0
0.1
0.2
−2
−0.2
(a)
x
−0.1
0
0.1
0.2
0.1
0.2
x
−4
x 10
0.6
0.5
1
Boltzmann
conventional
0.4
Fx − F(0)
x
0
u
0.3
0.2
−1
0.1
Boltzmann
Navier−Stokes
−Fourier
0
−0.1
−0.2
(b)
−0.1
0
0.1
0.2
(b)
x
Boltzmann
conventional
0.2
p
−0.1
0
x
Figure 4: Comparison of (a) the viscous stress, and (b)
the diffusive energy flux, for the same problem. The nonequilibrium parts of Πxx and Fx are shown solid, and
the Navier–Stokes–Fourier values for these quantities computed from the numerical values of ρ, u, θ are shown dotted. There is reasonable agreement except near the contact
discontinuity (x ≈ 0.08), where the true solution should
have spatially uniform velocity.
0.3
0.25
−2
−0.2
0.15
0.1
place no restriction on the flows that may be simulated.
0.05
(c)
0
−0.2
−0.1
0
0.1
0.2
x
Figure 3: Comparison of (a) the density, (b) the velocity,
and (c) the pressure for Sod’s first shock tube problem with
γ = 7/5 at time t = 0.2. The pressure and temperature
have been scaled by a factor of 1/4 to bring them inside the
stability window of the discrete Boltzmann scheme, which
has the effect of halving the velocity. Results from the
unsplit Boltzmann scheme are shown solid, and a conventional numerical solution of the inviscid problem is shown
dotted.
When extending to two or more spatial dimensions, the
number of degrees of freedom needed to satisfy the moment
constraints is typically few than the number needed for
isotropy. For example, in two dimensions the split model
requires 10 degrees of freedom for the momentum equation,
with distribution functions fi , and another 6 for the energy equation with distribution functions gi . Existing lattice Boltzmann models [21, 38] using the so-called D2Q13
square lattice could be adapted to simulate the density and
momentum equations in the split model. The second discrete Boltzmann equation for the gi could be implemented
on the common D2Q9 lattice, as in the Boussinesq-like
thermal lattice Boltzmann model of He et al. [13]. This
leaves 6 degrees of freedom that could in principle be cho-
12
1
editor, Computational Fluid and Solid Mechanics, pages
632–635, Amsterdam. Proceedings of The Third MIT
Conference on Computational Fluid and Solid Mechanics, Elsevier.
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unsplit 7 velocity
0.8
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general equations of state in lattice Boltzmann equations. Physica A, 362:132–138.
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0.6
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0.2
0
−0.2
−0.1
0
0.1
[12] Hanson, F. B. and Morse, T. F. (1967). Kinetic
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x
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